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The $ p$-weak gradient depends on $ p$


Authors: Simone Di Marino and Gareth Speight
Journal: Proc. Amer. Math. Soc. 143 (2015), 5239-5252
MSC (2010): Primary 46G05, 49J52, 30L99
DOI: https://doi.org/10.1090/S0002-9939-2015-12641-X
Published electronically: April 2, 2015
MathSciNet review: 3411142
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Abstract: Given $ \alpha >0$, we construct a weighted Lebesgue measure on $ \mathbb{R}^{n}$ for which the family of nonconstant curves has $ p$-modulus zero for $ p\leq 1+\alpha $ but the weight is a Muckenhoupt $ A_p$ weight for $ p>1+\alpha $. In particular, the $ p$-weak gradient is trivial for small $ p$ but nontrivial for large $ p$. This answers an open question posed by several authors. We also give a full description of the $ p$-weak gradient for any locally finite Borel measure on $ \mathbb{R}$.


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Additional Information

Simone Di Marino
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay Cedex, France
Email: simone.dimarino@sns.it

Gareth Speight
Affiliation: Scuola Normale Superiore, Piazza Dei Cavalieri 7, 56126 Pisa, Italy
Email: gareth.speight@sns.it

DOI: https://doi.org/10.1090/S0002-9939-2015-12641-X
Received by editor(s): December 4, 2013
Received by editor(s) in revised form: July 22, 2014, and August 27, 2014
Published electronically: April 2, 2015
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society